Is it.......

Prove that all numbers can be expressed with 2 and 5, or give a reason why it is impossible.

#NumberTheory

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

What exactly do you mean by "can be expressed with 2 and 5" ?

Prasun Biswas - 6 years ago

e.g. 3.5 = $2^{2}-2^{0}+2^{-1}$ , and 85.54 = $2^{4}(2^{2}+1)+5^{1}+2^{-1}+5^{-2}$

Bryan Lee Shi Yang - 6 years ago

I don't think $3.5$ and $85.54$ are integers, but I get your point.

Prasun Biswas - 6 years ago

You have my algorithm

Agnishom Chattopadhyay - 6 years ago

I think that it should be that every "terminating decimal number" can be expressed with 5 and 2, because are base is 10.

Archit Boobna - 6 years ago

Proof:

The following program expresses any positive integer in terms of 2 and 5

n = int(raw_input())
buffer = ""

for i in xrange(n-1):
   buffer += "(5-(2*2)) +"

buffer += (5-(2*2))
print buffer

The proof of correctness has been left to the reader as an exercise

Agnishom Chattopadhyay - 6 years ago

[This is just a mathematical explanation for the the algorithm given by Agnishom]

First, note that,

$0=5\times 2-2\times 5$

Now, consider any positive integer $x$ . We then have,

$x=0+x=(5\times 2-2\times 5)+\sum_{k=1}^x 1=(5\times 2-2\times 5)+\sum_{k=1}^x (5-2\times 2)\\ \implies x=(5\times 2-2\times 5)+\underbrace{(5-2\times 2)+(5-2\times 2)+\ldots}_{x\textrm{ times}}$

This is obviously a representation in $2$ and $5$ , as the OP wants.

It suffices to show that there is such a representation also for $(-x)$ to complete the proof. Indeed, we just multiply the above result by $(-1)$ to get,

$(-x)=(2\times 5-5\times 2)+\underbrace{(2\times 2-5)+(2\times 2-5)+\ldots}_{x\textrm{ times}}$

Since there exists such representations for $x~,~(-x)$ and $0$ where $x\in\Bbb{Z^+}$ , the claim in the problem is proved.

This is kinda similar to an inductive proof.

Prasun Biswas - 6 years ago

Hint: $2 \times 5 = 10$

Bryan Lee Shi Yang - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$